13.14.7 problem 7

Internal problem ID [2447]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number : 7
Date solved : Sunday, March 30, 2025 at 12:01:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 t^{2} y^{\prime \prime }+3 t y^{\prime }-\left (1+t \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 48

Order:=6; 
ode:=2*t^2*diff(diff(y(t),t),t)+3*t*diff(y(t),t)-(t+1)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \frac {c_2 \,t^{{3}/{2}} \left (1+\frac {1}{5} t +\frac {1}{70} t^{2}+\frac {1}{1890} t^{3}+\frac {1}{83160} t^{4}+\frac {1}{5405400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_1 \left (1-t -\frac {1}{2} t^{2}-\frac {1}{18} t^{3}-\frac {1}{360} t^{4}-\frac {1}{12600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )}{t} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 86

ode=2*t^2*D[y[t],{t,2}]+3*t*D[y[t],t]-(1+t)*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \sqrt {t} \left (\frac {t^5}{5405400}+\frac {t^4}{83160}+\frac {t^3}{1890}+\frac {t^2}{70}+\frac {t}{5}+1\right )+\frac {c_2 \left (-\frac {t^5}{12600}-\frac {t^4}{360}-\frac {t^3}{18}-\frac {t^2}{2}-t+1\right )}{t} \]

✓ Sympy. Time used: 0.965 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*t**2*Derivative(y(t), (t, 2)) + 3*t*Derivative(y(t), t) - (t + 1)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \sqrt {t} \left (\frac {t^{4}}{83160} + \frac {t^{3}}{1890} + \frac {t^{2}}{70} + \frac {t}{5} + 1\right ) + \frac {C_{1} \left (- \frac {t^{6}}{680400} - \frac {t^{5}}{12600} - \frac {t^{4}}{360} - \frac {t^{3}}{18} - \frac {t^{2}}{2} - t + 1\right )}{t} + O\left (t^{6}\right ) \]